armature copy rotation weirdly flips the direction
After researching this issue, I get the impression that there is something problematic about how Blender calculates rotations internally which makes rotation constraints reset bone values to zero whenever exceeding +/-180 degrees. It turns out that Blender continues calculating rotations even within ranges for which constraints are currently limiting rotations, and that these internal rotation calculations can continue after passing the constraint thresholds. What I find particularly baffling is that unchecking the For Transform checkbox for a Limit Rotation constraint stops "out of bounds" rotation values from being reported in the Transform panel, yet this doesn't actually stop the calculations themselves, which will still go on to create flips when transforming past +/- 180 degrees. This leads me to wonder what changing this setting meaningfully, operationally achieves, because it doesn't do anything to change this insufferable behavior. 

I noticed in the video that the author doesn't seem to have your issue. But then again, I don't think he ever rotated the control bone far enough at any point to tell me for sure that his rig wouldn't have done what yours is doing. So it's not clear that his rig would behave differently than yours if pushed past +/- 180 degrees. 

Some elsewhere have suggested creating drivers to address this problem, which seems to me like a very labor intensive workaround for something that shouldn't even be a problem. I noticed that the author creates drivers toward the end of the video, and I don't see any drivers in your file. I don't know that adding these drivers would have solved your issue, as I only skimmed the video. But I found it of possible interest that you didn't have any drivers set up before reporting this issue. 

I can't find any problems with the rig itself, but I could have missed something. I have found other links to similar questions which either go without answers or explicitly report that there is no solution.

I would really like to see a satisfactory solution presented, personally.